Hopefully competition will be more along technological lines and in the form of technical innovation, which all can benefit from. For instance, imagine if SpaceX and Blue Origin get into an "arms race" on building bigger and more capable rockets - it could lead to radical improvements on space launch capability and on the cost of access to space.
There was a popular videogame series called "Red Faction" which was about Martian miners rebelling against greedy corporate mining overlords from Earth. Hopefully, competition won't lead us towards that kind of unpleasant future - exactly the kind that Heinlein would have written about:
Heinlein sometimes depicted a harsher view of existence in outer space, analogous to previous human endeavors like the colonization of the Americas. Stories of his I remember reading as a child were: Starship Troopers, The Moon is a Harsh Mistress, Rocketship Galileo, Stranger in a Strange Land, The Puppet Masters, Have Spacesuit Will Travel, Starman Jones, The Starbeast, Between Planets, Time for the Stars, Podkayne of Mars, Space Cadet.
Hopefully, space industry innovation won't just be in launch vehicles, but all the other areas related to maintaining quality of life - particularly in robotics, which will be our essential workforce in space - and probably increasingly on Earth as well.